In three-dimensional space, points are represented by their positions along the -, -, and -axes, which are each perpendicular to one another; this is analogous to the 2d coordinate geometry interpretation in which each point is represented by only two coordinates (along the - and -axes).
In the figure above, the goal is to find the distance from the point to the point From the distance formula in two dimensions, the length of the the yellow line is
Let be the distance from the point to (the red line, and the desired distance). Then, using the Pythagorean theorem,
The above equation is the general form of the distance formula in 3D space. A special case is when the initial point is at the origin, which reduces the distance formula to the form
where is the terminal point. This equation extends the distance formula to 3D space.
Find the distance between the points and
From the distance formula, we have
How far is the point from the origin?
Since the second point is the origin, or , the distance is
If the distance between the two points and is 8, what is the value of
From the distance formula,
Determining the distance between a point and a plane follows a similar strategy to determining the distance between a point and a line. Consider a plane defined by the equation
and a point in space. Then the normal vector to the plane is
and the vector from an arbitrary point on the plane to the point is
The distance from the point to the plane is the projection from onto , or
The strategy behind determining the distance between 2 skew lines is to find two parallel planes passing through each line; this is because the distance between two planes is easy to calculate using vector projection. Furthermore, the normal vector to these 2 planes can be calculated using the cross product of the vectors representing the direction of the two lines.
In particular, suppose the two lines travel in the directions
then the normal to the planes can be calculated as
and the distance between the two planes--which is the same as the distance between the two lines--can be calculated by projecting onto , where and are points on the first and second lines, respectively.
Find the distance between the lines
Following the above strategy, the first line travels in the direction
and the second in the direction
Since and are points on the two lines, respectively, the vector is . Then